Potential, double-well

As the approach utilizes internal coordinates, and requires the calculation of the effective ID potential (double well potential in the case of the inversion motion of ammonia) two problems need to be solved beforehand a) the selection of the LAM... 

Figure Al.1.5. Ground state wavefimetion of the double-well oseillator, as obtained in a variational ealeulation usmg eight basis funetions eentred at the origin. Note the spurious oseillatory behaviour near the origin and the loeation of the peak maxima, both of whieh are well inside the potential minima.

The decrease in reactivity with increasing temperature is due to the fact that many low-energy ion-molecule reactions proceed tln-ough a double-well potential with the following mechanism [82] ... 

Figure B3.4.10. Schematic figure of a ID double-well potential surface. The reaction probabilities exliibit peaks whenever the collision energy matches the energy of the resonances, which are here the quasi-bound states in the well (with their energy indicated). Note that the peaks become wider for the higher energy resonances—the high-energy resonance here is less bound and Teaks more toward the asymptote than do the low-energy ones.

Figure B3.6.3. Sketch of the coarse-grained description of a binary blend in contact with a wall, (a) Composition profile at the wall, (b) Effective interaction g(l) between the interface and the wall. The different potentials correspond to complete wettmg, a first-order wetting transition and the non-wet state (from above to below). In case of a second-order transition there is no double-well structure close to the transition, but g(l) exhibits a single minimum which moves to larger distances as the wetting transition temperature is approached from below, (c) Temperature dependence of the thickness / of the enriclnnent layer at the wall. The jump of the layer thickness indicates a first-order wetting transition. In the case of a conthuious transition the layer thickness would diverge continuously upon approaching from below.

Fig. 2. The fluctuating difference between the proton potential at the product side relative to that at the reactant side (the difference between the two wells in a double-well proton potential). Whenever this difference is close to zero, tunneling conditions are favourable.

Figure 1 Double well potential for a generic conformational transition showing the regions of reactant and product states separated by the transition state surface.

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

The contour plot is given in fig. 43. As remarked by Miller [1983], the existence of more than one transition states and, therefore, the bifurcation of the reaction path, is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of this PES has been carried out by Benderskii et al. [1991b]. The... 

As seen from this table, the WKB approximation is reasonably accurate even for very shallow potentials. At 7 = 0 the hindered rotation is a coherent tunneling process like that studied in section 2.3 for the double well. If, for instance, the system is initially prepared in one of the wells, say, with cp = 0, then the probability to find it in one of the other wells is P( jn, t) = 5sin (2Ar), while the survival probability equals 1 — sin ( Ar). The transition amplitude A t), defined as P( + t) = A t), is connected with the tunneling frequency by... 

We hope to have convinced the reader by now that the tunneling centers in glasses are complicated objects that would have to be described using an enormously big Hilbert space, currently beyond our computational capacity. This multilevel character can be anticipated coming from the low-temperature perspective in Lubchenko and Wolynes [4]. Indeed, if a defect has at least two alternative states between which it can tunnel, this system is at least as complex as a double-well potential—clearly a multilevel system, reducing to a TLS at the lowest temperatures. Deviations from a simple two-level behavior have been seen directly in single-molecule experiments [105]. In order to predict the energies at which this multilevel behavior would be exhibited, we first estimate the domain wall mass. Obviously, the total mass of all the atoms in the droplet... 

Needless to say, tunneling is one of the most famous quantum mechanical effects. Theory of multidimensional tunneling, however, has not yet been completed. As is well known, in chemical dynamics there are the following three kinds of problems (1) energy splitting due to tunneling in symmetric double-well potential, (2) predissociation of metastable state through... 

The next debate in the literature was whether these molecules have C2v or Cs symmetry. The nuclear motion of a C2v symmetric structure would be described by a single-well potential (see Figure 10). The alternative is a rapid interconversion of two valence tautomers, each of Cs symmetry. This would occur via the C2v structure as transition state (see Figure 11). In this case the motion of the central sulfur would be described by a double-well potential, and dioxathiapentalene and trithiapentalene would be misnomers for (3//-l,2-oxathiol-3-ylidene)acetaldehyde 180 and (3/7-1,2-dithiol-3-ylidene)thioacetaldehyde 181. One advantage of C2v symmetry is aromatic stabilization from the 1071 electrons <2001CRV1247>. The alternative Cs symmetry has the advantage of avoiding a hypervalent sulfur. 

Fig. 8.3. Histogram of work values for Jarzynski s identity applied to the double-well potential, V(x) = x2(x — a)2 + x, with harmonic guide Vpun(x, t) = k(x — vt)2/2, pulled with velocity v. Using skewed momenta, we can alter the work distribution to include more low-work trajectories. Langevin dynamics on Vtot(x(t),t) = V(x(t)) + Upuii(x(t)yt) with JcbT = 1, k = 100, was run with step size At = 0.001, and friction constant 7 = 0.2 (in arbitrary units). We choose v = 4 and a = 4, so that the barrier height is many times feT and the pulling speed far from reversible. Trajectories were run for a duration t = 1000. Work histograms for 10,000 trajectories, for both equilibrium (Maxwell) initial momenta, with zero average and unit variance, and a skewed distribution with zero average and a variance of 16.0...

As an illustrative example of this approach, consider applying Jarzynski s identity to reconstruct a one-dimensional double-well energy profile of the form V(x), assumed unknown and which is to be recovered by the method, from pulling trajectories with a harmonic guiding potential... 

The gradient-squared term in the above equation represents the energy of the interface separating the phases the constant C can be interpreted as a measure of the interaction range. The bulk potential /(< )) has the Landau-Ginzburg (GL) 4>4 (double-well) structure... 

Figure 5 Double-well potential function constructed to fit the vibrational spectrum of ammonia.

Fig. 1.11. Ground state and excited charge resonant states of a model doubly-charged diatomic molecule. The electronic configurations of the states are shown schematically in a 1-D double-well potential...

Increasing a leads to the effective double-well potential shown earlier with two elliptic (stable) and one hyperbolic (unstable) fixed points. The elliptic fixed points become unstable for parameter values below... 

It is noted that st-20 also has a similar double-well potential curve at about P 160° and M 200° dihedral angles and the global minimum M is slightly... 

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